What is the differect between them?
jquery-ui-1.7.3.zip: is the basic version, no additional theme
jquery-ui-1.7.3.custom.zip: is a custom version, you must choose which components and additional theme. After click Download then a custom jquery created for you. REMEMBER that in this "custom" build there is a new folder called "development-bundle"

jquery-ui-1.7.3.zip: if you decide to use jQuery UI basic version then upload jquery-ui-1.7.3.zip to /sites/all/modules/jquery_ui, after extract this file you will have jquery-ui-1.7.3 folder, you must rename to jquery.ui, so your folder become: /sites/all/modules/jquery_ui/jquery.ui
Remember: jquery dot ui (jquery.ui) not jquery_ui !

jquery-ui-1.7.3.custom.zip: if you decide to use jQuery UI custom version then upload jquery-ui-1.7.3.custom.zip to /sites/all/modules/jquery_ui, after extract this file you will have jquery-ui-1.7.3 folder also, you must rename to jquery.ui also, so your folder become: /sites/all/modules/jquery_ui/jquery.ui
BUT .... you need additional step: open "development-bundle" folder and move all contents of /sites/all/modules/jquery_ui/jquery.ui/development-bundle to /sites/all/modules/jquery_ui/jquery.ui, you then may delete this empty development-bundle folder

The reason (as far as I can tell) why 1.8.10 doesn't work is because in that version the names of the script files has been changed to jquery.ui.widgets.min.js, etc. so the links to the files are all wrong.

If you've just downloaded 1.7.3 or 1.8.x from http://jqueryui.com/download, the jquery.ui folder might not have sufficient permissions by default, which will result in a "jQuery UI plugin is missing..." error on the modules page. "chmod 755 jquery.ui" worked for me.